Electron. J. Differential Equations, Vol. 2023 (2023), No. 53, pp. 1-17.

Optimal energy decay rates for viscoelastic wave equations with nonlinearity of variable exponent

Muhammad I. Mustafa

In this article, we consider the viscoelastic wave equation $$ u_{tt}-\Delta u+\int_0^{t}g(t-s)\Delta u(s)ds+a| u_t| ^{m(.)-2}u_t=0 $$ with a nonlinear feedback having a variable exponent m(x). We investigate the interaction between the two types of damping and establish an optimal decay result under very general assumptions on the relaxation function g. We construct explicit formulae which provide faster energy decay rates than the ones already existing in the literature.

Submitted March 22, 2023. Published August 28, 2023.
Math Subject Classifications: 35B40, 74D99, 93D15, 93D20.
Key Words: Viscoelasticity; frictional damping; variable exponent; energy decay.
DOI: 10.58997/ejde.2023.53

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Muhammad I. Mustafa
Department of Mathematics
University of Sharjah, P.O. Box 27272
Sharjah, United Arab Emirates
email: mmustafa@sharjah.ac.ae

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